Fully nonlinear second order elliptic equations : recent development

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

N ICOLAI V. K RYLOV

Fully nonlinear second order elliptic equations : recent development

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 25, n

^o

3-4 (1997), p. 569-595

<http://www.numdam.org/item?id=ASNSP_1997_4_25_3-4_569_0>

© Scuola Normale Superiore, Pisa, 1997, tous droits réservés.

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569

Fully Nonlinear Second Order Elliptic Equations:

Recent Development

NICOLAI V. KRYLOV

Abstract. A short discussion of the history ^{of the} theory ^of fully ^{nonlinear second-}

order elliptic equations ^is presented starting ^{with the} beginning ^{of the} century.

Then an account of the explosion ^{of results} during ^the last decade is given. ^This explosion ^{is based} entirely ^{on a} generalization ^for nondivergence form linear operators of the celebrated De Giorgi ^result bearing ^{an Holder} continuity. ^{This is}

an extended version of a 1.5 hour talk at Mathfest, Burlington, Vermont, Aug ^6- 8, 1995.

Mathematics Subject Classification (1991): ^35J60.

1. - Introduction

It seems Bernstein

[8]10

^{in 1910 was the first to introduce}

general

^methods

of

solving

^nonlinear

elliptic equations.

^{These are:} the method of

continuity

^and

the method of a

priori

estimates. He considered

equations

^{with two}

independent

variables and showed that for

proving

^the

solvability

^{of such}

equations

it suffices

to establish a

priori

estimates for absolute values of the first two derivatives of solutions. For Bernstein the

equation

where Q

is

^{a domain in}

^R~,

should be called

elliptic

^if

the

^matrix

^{(F~l~ )}

^is

definite. This definition is

quite

natural when the

equation

^{is linear with} respect

to second order

derivatives,

^{but is not of much}

help

^for

fully

^nonlinear

equations.

For

example,

before 1910 in 1903 Minkowski

[65]°3 proved

^{existence and}

uniqueness

^{of convex} surface with

prescribed

^{Gaussian curvature in Euclidean}

space. He did not prove however that this surface is from class

C2.

Analytically,

the Minkowski

problem

^involves

solving

^a

highly

^nonlinear

partial

differential

equation

^of

Monge-Ampere

type. ^An

example

^{of such an}

This work was supported ⁱⁿ part by ^NSF grant DMS-9302516.

(1997), pp. 569-595

equation,

^{for a convex function}

u (x) - u (x 1, ... , xd )

^{defined in a domain in}

R d

is the

following simplest Monge-Ampere equation

where

f (x )

^{is a}

given

function. This

equation

^{is not}

elliptic

^{in the sense of}

Bernstein.

It turns out that

(1.2)

and similar

equations

^{can be} well understood even for

non differentiable

functions,

^{so that one can}

investigate generalized

^solutions.

This was done

by

Aleksandrov

[ 1 ]5g

ⁱⁿ 1958 and led to some remarkable results for linear

equation

^{which 20} years later turned out to constitute the basis of the

general theory

^of

fully

^nonlinear

elliptic equations.

^{We mean} the so-called Aleksandrov maximum

principle

and Aleksandrov estimates

(see [2]61

^and

[3]63).

The smoothness of Minkowski’s two-dimensional

generalized

^{solution was}

studied

by

well-known

mathematicians,

^{such as}

Lewy [59]3~,

^Miranda

[66]39,

⁹

Pogorelov [69]52, Nirenberg [67]53,

^Calabi

[19]58 ,

^Bakel’man

[6]65.

^{In 1971}

that is 68 years after Minkowski’s

work, Pogorelov

ⁱⁿ

[71 ] ~

¹

finally proved

^that

the solution is indeed from class

C2

in multidimensional case. Nevertheless his

proofs

ⁱⁿ

[71]71

^{1 and}

[72]~5

^{contained what looked like a} vicious circle and

only

in 1977

Cheng

^{and Yau}

[2 1]77

^{showed how to avoid it.}

To prove the existence of solutions of

equations

^like

(1.2) by

^{the methods}

known before 1981 was no easy task. It involved

finding

^a

priori

^estimates

for solutions and their derivatives _up to the third order.

Big part

of the work is based on

differentiating (1.2)

^three times and on certain

extremely cleverly organized manipulations

^invented

by

^Calabi. After 1981 the

approach

^to

fully

nonlinear

equations changed dramatically.

^We discuss this in Section 3.

Until 1971 the

theory

^of

fully

^nonlinear

elliptic equations only

^{consisted of}

the

theory of Monge-Ampere equations.

^{In 1971}

appeared

^the so-called Bellman

equations

which stemmed from the

theory

of controlled diffusion processes. The

typical representative

of Bellman’s

equations

^{is the}

following:

where A is a set and

(aij)

⁼

(a’j)* &#x3E;

^0.

To non

specialists equations

^{in form}

(1.3) might

look artificial.

Indeed, equations

which arise in other fields of mathematics look different. For

example,

in many natural

geometrical applications

^{there appear}

equations

^{like the}

Monge- Amp6re equations

^and

prescribed

^curvature

equations

or more

general equations

where

Hyn

denotes the m-th

symmetric polynomial

^{of the curvature matrix of}

the

graph

^{of u.}

Equations

of the form of the

complex Monge-Ampere equation:

are also _very

popular

^{in the} literature. These

equations

^{arise in}

complex

geom- etry ^and

complex analysis.

^The

following equation

where I is the d x d unit

matrix,

^{comes from the}

theory

of calibrated

geometries (see Caffarelli, Nirenberg

^and

Spruck [16]85)

in connection with

absolutely volume-minimizing

submanifolds of

R~.

Its

particular

^case

^(when

^{d =}

3)

was considered earlier

by Pogorelov [70]6°

^{in two} dimensions and

by

^{the au-}

thor

[42] g3 .

^{It is worth}

noting

^that

although equations ( 1.4)-( 1.7)

look different from the Bellman

equation ^(1.3),

nevertheless each of them is

equivalent

^{to a}

corresponding

^Bellman

equation

^of type

(1.3).

^With

regard

^to

(1.4)

^and

(1.6)

this fact was known in

1971,

^{which made the}

theory

of Bellman’s

equations

very attractive.

Really general theory

^of

fully

^nonlinear

elliptic equations emerged

from the

theory

of Bellman’s

equations.

^{We will see later that even now the}

most

general theory

^of

fully

^nonlinear

elliptic equations,

ⁱⁿ

^fact,

^{reduces to the}

theory

of Bellman’s

equations.

At

early stages

^{of the}

theory

^{of Bellman’s}

equations

^the

only

^available

methods were those of the

theory

^of

probability.

^It is remarkable that results obtained

by

these methods are

sharp

in many

situations,

^and even now some of

them,

which admit

purely analytic formulation,

^are

only

^obtained

by probabilistic

means. The reader can make an

acquaintance

^{with the}

corresponding

^results

starting

^{with the book}

by Fleming

^{and Sooner}

[27]93

and with references therein

to which we

only

^add

Pragarauskas [73]82

^and

Krylov [46]g9.

By

^the way for ^one of discontinuous controlled processes considered in

[73] 82

Bellman’s

equation

takes the

following

^form:

Until 1982 the

probabilistic

^{methods were the most}

powerful

^{in the}

general theory

^of

fully

^nonlinear

elliptic equations.

^{The situation}

changed dramatically

in 1982 when Evans

[25]g2

^{and the author}

[40]g2 proved

^the

solvability

ⁱⁿ

c2+a of a

broad class of Bellman’s

elliptic

^and

parabolic equations.

^The

proofs

were based on the

theory

^{of linear}

equations and,

ⁱⁿ

particular,

^{on the fact}

that one can estimate the Holder constant for solutions of linear

equations

^with

measurable coefficients. The latter fact was

previously

established in 1979

by

Safonov and the author

[52]~9 (see

^also

[53]g°

^{and a beautiful}

exposition by

Safonov for

elliptic

^{case in}

[75]80) ,

^as mentioned before on the sole basis of Aleksandrov’s estimates.

Remarkably enough, although

^the

proofs

ⁱⁿ

[53]g°

and

[75]80

^{are written in PDE} terms, all

underlying

^{ideas are}

probabilistic,

^and

perhaps

^{this was the reason}

why people

^{from PDEs did not succeed in}

obtaining

the estimate before. After this the

general

^PDE

theory

^of

fully

^nonlinear

elliptic equations

^started up, and below we will

give

^a

report

^on

major developments

of this

theory.

The article is

organized

^{as follows. In Section 2 we}

give

^a

general

^notion

of

fully

^nonlinear

elliptic equation

^{and some} existence theorems. Section 3 is devoted to results

bearing

^{on the}

general theory

^of

fully

^nonlinear

uniformly elliptic equations

and Section 4 contains a discussion of results for the

general theory

^of

fully

^nonlinear

degenerate elliptic equations.

^In Section 5 we

speak

about

equations

^{related to the}

Monge-Ampere equations.

Section 6 contains

a new

(and probably

^the

^first)

^{result on the rate of} convergence of numerical

approximations

^for

fully

^nonlinear

degenerate elliptic equations.

^{We have}

already

mentioned above that the modem

theory

^of

fully

^nonlinear

elliptic equations

is based

entirely

^{on some}

deep

results from linear

theory.

Also many ^new brilliant ideas and

techniques appeared.

^{In Section 7 we} present ^{one of them} which is Safonov’s

proof

^{of the} Holder-Korn-Lichtenstein-Giraud estimate for the

Laplacian.

^This

proof

^was

designed

^{for nonlinear}

equations

^{and turned out}

to be shorter and easier than usual ones even for the

simplest

^linear

equation.

^In

my

opinion

^his

proof

^{should be} part ^of

general

mathematical education.

Finally,

not as brilliant and not a very

popular

^{technical idea is}

presented

^{in Section 8.}

Exploiting

^{this idea} allowed the author to

get

^some very

general

^{results on}

fully

nonlinear

degenerate elliptic equations.

^{Also the idea is of a}

general

^character

and

might

be of interest to mathematicians from other areas.

It is to be said that the literature on

fully

^nonlinear

elliptic equations

^is

really ^immense,

^we present ^{here a} report ^{on the}

only

part of it which is close

to interests of the author. In

particular,

^{we do not discuss concrete}

applications

in which

equations

^{we discuss arose.}

2. - A

general

^{notion of}

fully

^nonlinear

elliptic equation

^and

examples

Conceivably,

the first

question

which arises when a

theory

^{starts is: what}

is the main

object

^of

investigation? Interestingly enough

^{this was not the first}

question

^{addressed in the case of the}

theory

^of

fully

^nonlinear

elliptic equations.

The reason for this is that there were

enough

^old

problems regarding fully

nonlinear

equations

^{which came} up earlier and

they

^{were to} be solved in the first

place.

Now when the

general theory

is rather well

developed,

^one may think how

to make the field of its

applications

^{as wide as}

possible

^{and the number of}

people

^{who can use it as}

large

^as

possible.

We have the

following

situation. From the one

hand,

^a

huge variety

^of

results is available in the

theory.

^{On the other hand}

however,

^{it turns out that if}

an

inexperienced

^{reader meets a}

fully

nonlinear second order

partial

differential

equation

^{in his}

investigations

^{and tries to get any} information

concerning

^its

solvability

^{from the}

literature,

then almost

certainly

^{he fails to} find what he

needs,

^{unless he considers an}

equation

^{that is}

exactly

^{one which had}

already

been treated. The

point

^{is that in the}

general theory

^{we treat}

only equations

which

satisfy

certain conditions and while

considering examples,

^{we show how}

to

transform

^the

equations

^{in these}

examples

^{to other}

equations

^{to which the}

theory

^is

applicable. Therefore,

^{from the}

point

^{of view of}

applications

^{the main}

question

^{is how to describe in}

simple

^{terms the most}

general

^{situation when one}

can make an

appropriate

transformation. In other terms, ^one needs a

general

notion of

fully

^nonlinear

elliptic equation.

Naturally,

^the type ^of

equation

should be defined

only by

^the way of de-

pendence

^{of F on}

D2 u,

^{that is we call our}

equation (1.1) elliptic

^{if for}

any p ^E y E S2 and z E R the

following equation

^{in Q is}

elliptic:

F(D2u(x),

p, Z,

y)

^{= 0.}

Therefore,

^{we have} to concentrate on the case when F

depends only

^{on the matrix of second order} derivatives of u, in other

words,

we have to consider the

equation

We assume of course that

Usually

ⁱⁿ the literature on nonlinear

elliptic equations (see,

^for

instance,

[22]62, [29]83, [14]g4, [15]gs, [44]g5)

^one accepts the definition

by

^Bernstein

and

equation (2.8)

is called

elliptic

^{if the matrix is}

nonnegative (or nonpositive)

^{for all} arguments. ^{As we have noticed}

above,

^{this excludes at}

once even the

simplest Monge-Ampere equation

since for

F(uij)

^:=

the matrix is definite if and

only

^{if the same is true for}

(uij).

An attempt ^to

give

^a better definition is made in

[6]65

^{where the}

equation

is called

elliptic

^{on a}

given

^{solution u if at} any

point

^{x E Q the matrix with}

entries

(D2u (x))

^is

nonnegative (or nonpositive).

^{After that}

equation (2.8)

is called

elliptic

^{in a}

given

^{class C of functions}

(say,

^{C is}

C2 (S2)

^{or the set of}

all smooth convex

functions)

^{if it is}

elliptic

^on any

(if

^{there is}

any)

^solution

u E C. It is worth

noting

^that

only

^{in rare} cases we can take C ⁼ in this definition. For

instance,

^{as we have seen}

above,

^{this is not}

possible

^{for the}

Monge-Ampere equation. ^However,

^the

Monge-Ampere equation

^is

elliptic

^on

convex functions. But how to find an

appropriate

^{C for the}

following equations:

If we are

only

interested in definiteness of

(7~..),

^{then as easy to}

^check,

equations (2.9), (2.10)

^{are both}

elliptic

^{in the same class of} functions C defined

as the set of all functions for which

1/,JÏ8.

^{It turns out that in}

general

the Dirichlet

problem

^for

^(2.9)

is solvable in this class and for

(2.10)

is not, and

moreover the behavior of solutions of

(2.10)

^is such that this

equation

^should

not be called

elliptic

^{at all.}

Other flaws of the definition are also related to the fact that the

objective

is not

only

^to

give

^a definition of nonlinear

elliptic equation,

^{but to find such}

a definition which could do the

job.

^For

instance, usually

^{we are} interested in

proving uniqueness,

^and

usually

^we prove it via the maximum

principle.

^In

other

words,

^{if we are}

given

^{two solutions ui, u2 of}

equation (2.8),

^then

by proceeding

^{as usual}

(cf.,

^for

instance, [22]62

^Ch.

4,

^Section

6.2)

^{for v =} u 1- u2 we write

where

and we expect ^{the matrix a = to be}

positive

^or

negative.

^{If we assume}

the above definition from

[6]65,

^{then we} know that the matrices are say,

positive

^{on u 1 and on u 2, but}

generally speaking,

^{tul 1 -f-}

(I - t ) u 2

^{is not a} solution and we do not know

anything

about definiteness of a.

Actually,

^it may

even

happen

^{that for one function F the} matrix a is

always positive,

^{and for}

another function

F, defining

^an

equation equivalent

^to the initial

equation (2.8),

the

corresponding

matrix a is neither

positive

^nor

negative.

^The

point

^{is that}

we can

arbitrarily modify

the function F outside the set

r,

^the

only

^{set where}

some

properties

^{of F are}

given

^{so far.}

By

^the way, this

possibility

^of

modifying

nonlinear

equations

is the main reason for the radical difference between linear and nonlinear

equations,

since for the linear case the set r is a

hyperplane

ⁱⁿ

the linear _space

where k ⁼ and there are not so many ways ^to

represent

^a

hyperplane

^as

null set of a linear function.

One _way to overcome the last

difficulty

^{is to}

accept

^the notion of

elliptic convexity

^{of F from}

[6]65,

^{that is to consider}

only

^F such that for _any two

solutions

(from

^{the class}

C)

the matrix a is

positive.

^{In this} system ^of

notions, given

^an

equation,

^to decide if it is a

"legal" elliptic equation,

^{we first should}

guess in what class of functions ^we will look for solutions and then to

modify (if

^{it is}

possible

^at

all)

the function

F,

^without

changing

^the

equation,

^{in order}

to

replace

^{it with an}

elliptically

^{convex F. For the}

Monge-Ampere equations

appropriate

modifications are

Unfortunately,

^even after this other difficulties still remain. For

instance,

assume that at the very

beginning

^{we know the}

appropriate

^class of functions

C,

^{and our F is}

elliptically

^{convex in} this class. Assume that we even obtained

a

priori

estimates for solutions of the

equation.

^The

question

^{arises how to}

prove existence theorems.

Usually

^{we introduce a} parameter t ^E

[0, 1]

^{and we} try ^to find functions

Ft

continuous in t such that

FI =

^{F and}

Fo

^{defines an}

equation

^{for which} every-

thing

^is known. After this we are

trying

^to prove the ^{same a}

priori

estimates for

solutions, belonging

^{to the same class}

C,

^{of the}

equations corresponding

^to

Ft

^for all t E

[0, 1],

^{and then we}

apply

^some

topological

^{methods to} get ^the

solvability

of the

equation

^{= 0 for t = 1 from its}

solvability

^{for t = 0. But on}

this _{way, in} all

interesting

^{cases, we cannot afford to take}

Fo

linear since

usually

solutions of linear

equations

^have no reasons to

belong

^{to C. For}

^instance,

^for

the

Monge-Ampere equation

^{det = 1 in a}

strictly

^convex domain S2 with

boundary

^{data on}

8Q,

^{one of the}

right

classes of solutions is the class of all

convex functions. At the same time there is no linear

equations

for which all solutions with different

boundary

^data are convex.

In a way, this cuts us off the linear

theory

and raises the obscure

problem

^of

finding

"model" nonlinear function

Fo

^for any

particular

^{F. For}

professionals

in the field this

problem

^{is not too}

hard,

^and many authors

prefer

^{to use model}

equations

^while

treating

^concrete

equations ^(see,

^for

^instance,

^Bakel’man

[6]65,

Caffarelli, Nirenberg

^and

Spruck [14]84,

^Ivochkina

[35]89 ).

^{But for a}

"ready-to-

use"

theory

this "cut off" is

highly

undesirable since

applications

may advance

equations

different from those which have

already

^been

investigated. ^However,

in the above system of notions we cannot avoid this

difficulty

^{unless we can}

either understand how to

modify

^{the method of}

continuity

^{in the situation when}

the set

Ct

of solutions is

evolving

^{with t, or we can} "hide" the set C

by finding

^a

function

F

^such that _any solution

of (2.8) of class

^{C is a solution to the}

equation

and vice _{versa, any} solution of

(2.12)

^{is a} solution of

(2.8)

^and

belongs

^{to C.}

Our definition is based on the latter

possibility.

Following [50]95

^{we shall} present ^{a different}

approach

^{to the notion of}

nonlinear

elliptic equation.

^{We shall}

give

^{a method to decide if a}

given

^nonlinear

equation

^{is an}

elliptic

^one

by looking only

^{at the}

equation

^without

using

any information

regarding

^the

problem

ⁱⁿ which this

equation appeared.

^{After this}

we

give

^a notion of admissible solutions of the

equation

^{and then we discuss}

the

possibility

^of

rewriting

^the

equation

^{with the}

help

^of

elliptically

^convex

functions F.

The most

important

concept ^{in our}

approach

^{is the} notion of admissible solutions which shows the

right

^{class of functions in which to look} for solutions.

This notion is based on the notion of

elliptic

branches of the

given equation,

which turns out to be

meaningful

^{even for}

viscosity

solutions of the

first

^order

nonlinear

equations.

It is worth

noting

^{that in all cases} known from the literature our class of admissible solutions coincides with the known ones.

Also,

^our notion has _many

common features with similar notions or

hypotheses

^from

Caffarelli, Nirenberg

and

Spruck [ 16] g5 , Trudinger [84]90.

Our

point

^{of view is based on the} observation that _every individual _equa- tion

(2.8)

^{means and means}

only

^{that for} any ^x E S2

This

point

of view allows us to concentrate on

properties

^{of the set r rather}

than occasional

properties

^{of numerous} functions which define the same set r.

Only properties

^{of the set r} define the type ^{of the}

equation.

Of course, we assume that F is at least a continuous

function,

^what

implies

that r is a closed set in the linear space

Sd.

We also

keep

^the

assumption

^that

r #

^0.

Finally,

^{remember that I is the unit d x d matrix.}

DEFINITION 2.1. We say that ^a nonempty open

(in

^set

e =1= Sd

^{is a}

(positive) elliptic

^{set if}

(a) @ = @ ) 8 (@),

(b)

for any ^E

30, ~

^E

R d

it holds that

(Uij +~~)

^{E 8.}

DEFINITION 2.2. _{We say} that

equation (2.8) (or,

^more

generally,

equa- tion

(2.13)

with any nonempty ^closed

r)

^{is an}

elliptic equation

^{if there is}

an

elliptic

^{set 0 such} that a0 C r. In this case we call the

equation

an

elliptic

^branch

of equation ^{(2.8) (or} (2.13)) defined bye.

Nonlinear

equations

may have many

elliptic

branches. For instance

(2.9)

has two and

(2.11)

^{has four}

elliptic

^branches.

DEFINITION 2.3. We say that ^an

elliptic

^{set 0 is}

quasi nondegenerate

^{if for}

any ^E

a O, ~

^E

R d B 101

^{we have -f-}

~’~i)

^{E 0.}

Given a number 3 &#x3E;

0,

^{we call an}

elliptic

^{set 6}

6-nondegenerate (or

^uni-

formly elliptic)

^{if for} any ^{W E}

a O ,

^E

Rdwe

have

If

equation (2.14)

^{is an}

elliptic

^{branch of}

(2.8) (or (2.13))

^{and 6 is}

quasi nondegenerate (8-nondegenerate, uniformly elliptic),

^we call this branch and

equation (2.8) (or (2.13))

^itself

quasi nondegenerate (respectively, 3-nondegener-

ate,

uniformly elliptic).

Notice that each of two

elliptic

branches of

(2.9)

^is

uniformly elliptic

^whereas

all branches of

(2.11)

^are

degenerate.

DEFINITION 2.4. Given an

elliptic equation (2.8) (or (2.13)),

^we say that ^a function u is an admissible solution in Q if u is a solution in Q of _any

elliptic

branch of the

equation ^(the

branch should be the same in the whole of

Q).

Note, ^for

instance,

^that

u(x, y) = x2 - y2

^{is not an} admissible solution of the

elliptic equation uxx u yy

^{= 16.}

The

following

^theorem shows that

equations

written in somewhat unusual form

(2.14)

^are

actually

^the

equations

^{which one treats in the}

general theory

^of

fully

^nonlinear

elliptic equations. Exactly

^{this theorem}

justifies

^our definition.

THEOREM 2.1. Let O be an

elliptic

^{set and}

equation (2.14)

^be

elliptic ( for

instance, be ^an

elliptic

^branch

of (2. 8) ). Define

Then

and in

particular, equation (2.12)

^is

equivalent

^to

equation (2.14). Furthermore, for any ~

^{E E}

^Sd

Moreover, the

function F

^is

elliptically

^{convex in the sense that}

for

any

(uij),

E

Sd

the

difference F (u ij) -

^{can be}

vij)

^with

a

nonnegative symmetric

^{matrix a.}

Finally, if equation (2.14)

^is

3-nondegenerate,

then

- - .. -

An immediate consequence of this theorem and of results from

Crandall,

Ishii and Lions

[23]92

^{is the}

following

THEOREM 2.2. Let S2 be a bounded smooth

domain, and 0

^{be a continuous}

function

^on a S2. Assume that

equation (2.8)

^{has a}

uniformly elliptic

branch. Then this

equation

^{with the}

boundary

^{condition u}

= 0

^{on a S2 has an} admissible

viscosity

solution u E

C (S2).

Moreover, every

uniformly elliptic

^branch

of (2. 8)

^{has its} _own

unique

admissible

viscosity

^{solution u E}

^C(Q).

°

One of the hardest and

exciting

open

problems

^{in the}

general theory

^of

fully

nonlinear

elliptic equations

^concerns smoothness of solutions when neither 6

nor its

complement

^{is convex.} If d &#x3E;

3, nothing

^{is known} about boundedness

or

continuity

^{of second order} derivatives of solutions. For

example, nothing

is known about classical

solvability

^{of the Dirichlet}

problem

^{for the}

following equation

where 1 k d. Theorem 2.2

only

says that the Dirichlet

problem

^is

uniquely

solvable in the class of

viscosity

^solutions.

Note that in Theorem 2.1 the

function F

^is

obviously

^{concave if 6 is convex,}

and it is convex if the

complement

^{of 0 is convex.}

Graphs

^of convex or concave

functions can be

represented

^as

envelopes

^{of their}

tangent planes.

^Therefore

equation (2.12)

^can be rewritten in the form of Bellman’s

equation (1.3).

^Ac-

tually,

^as easy ^{to see even} in

general

^case

equation (2.12)

^is

equivalent

^{to a}

Bellman

equation,

which contains sup and inf ^at the same time. If we combine this with results from

[44]85,

^{then we obtain}

THEOREM 2.3. Let Q be a bounded domain

of

^class

C2+,

where a E

(0, 1),

and

let 0

^E

C2+"

Assume that

equation (2.8)

^{has a}

uniformly elliptic

^branch

defined by

^{a domain 6 such that either 6 or its}

complement

^{is convex. Then this}

equation

^{with the}

boundary

^condition

u - ~

^{on a S2 has an} admissible solution

u E

c2+fJ (Q),

^where

f3

^E

(0, 1 ).

Moreover, ^the

elliptic

^branch

(2.14)

^{with the}

given boundary

condition has its own

unique

admissible solution u E

C2+0

This theorem

applies

^to

equation (2.9)

which has two

uniformly elliptic

branches.

General

theory

^from

[46]89

^or

[51 ]95

^also

implies

^the

following

^theorem

which can be restated in an obvious way for the ^case in which the

complement

of E) is convex.

THEOREM 2.4. Let E) be an open ^{convex set} and let

equation (2.14)

^be

elliptic.

Let C be an open ^cone in

Sd

with vertex at the

origin,

^{and let} to be ^a number. Assume that

to I

+ E) C

C,

^and

that for

any w E C ^we have tw E 0

for

^{all t}

large enough.

Let tr w &#x3E; 0

for

any

w E C,

^{and let Q be a}

strictly

^{convex domain}

of

^class

C4.

Then

for any 0

^{E there is a}

unique function

^{ú E}

C (0)

ⁿ

(Q)

^{such that}

u E ae

(a.e.)

^{in Q.}

If,

ⁱⁿ addition,

equation (2.14)

^is

quasi nondegenerate,

^{then u E}

C2+" (0) for

^{an a E}

(0, 1).

This is not the most

general

^{result from}

[46]89

^or

[51]95.

^{There we do not}

assume that Q is convex or that the weak

nondegeneracy

condition: tr w &#x3E; 0 for _any

w E C,

is satisfied. The assertion then becomes more

complicated.

The

general

^theorem

applies

^{to each of four}

elliptic

branches of

(2.11 ).

^For

the branch ⁼ 0 it _says that for _any

C4

function

given

^{on the}

boundary

^{of a}

^C4

^{domain Q C}

^JR2

there exists a

unique C1,I(Q)

solution with this

boundary

^value

provided

^{that the}

boundary

^{of Q is}

strictly

^convex

^(outward)

at any

point

^{where the}

tangent

^{line is}

parallel

^{to one} of the coordinate axis.

We say ^more about these results in Section 4.

To conclude this section we mention a

general

^{method how to}

perturb

^an

elliptic equation ^(2.8)

^{in order to} get ^a

uniformly nondegenerate

^{one. It suffices}

to take s &#x3E; 0 and consider

This method has been

constantly

^{in use in works}

by

the author.

Usually,

ⁱⁿ

the literature other methods are

applied.

^In connection with this it is worth

noticing

^{that the}

following

"natural"

perturbation AM + det D2U

^{= 1 of the}

Monge-Ampere equation ^{detd 2U}

^{= 1 is not}

elliptic

^{at all}

unless s s

^{0 !}

3. - General convex

fully

^nonlinear

uniformly elliptic equations

Since works of Bernstein it was known that to prove the

solvability

^{of dif-}

ferential

equations

^{it suffices to obtain a}

priori

^{estimates in}

appropriate

^classes

of functions for solutions under the

hypothesis

^{that the solutions exist. One}

of _ways to obtain estimates in is to differentiate the

equation

^{once thus}

obtaining

^a

quasilinear equation

^{or a} system ^of

equations

^with respect ^{to first} derivatives and

hope

that there is

Cl+"-estimate

for solutions of this new equa- tion. This

hope

^was

destroyed by

^Safonov

[77]87 .

^The

example

ⁱⁿ

[77]g~

^buried

the

hopes

^{for an}

"easy" theory

^of

fully

^nonlinear

equations,

^{and in a sense saved}

the work which has been done before for F’s either convex or concave with respect ^{to the matrix}

D 2u

and

sufficiently

^{smooth in} other variables. Below we

only speak

^{about such F’s.}

In works

by

^Evans

[25 ] g2

^{and the author}

[40]82

^we obtained interior a

priori

estimates. Then in

[41 ]83

^{the author}

published

^a

priori

^estimates up to the

boundary

^{for the Dirichlet}

problem.

After this _many authors con-

tributed to the

general theory

^of

uniformly elliptic equations.

^{We will}

only

mention works which

played

^{the most}

important

^{role in the}

development

^of

the

general theory.

^{We start} with works

by

^Evans

[26]83 , _Trudinger [8 1]83, Krylov [43]g4, Caffarelli, Kohn, Nirenberg

^and

Spruck [15]85, Caffarelli, Nirenberg, Spruck [ 16]gs .

The

major

results obtained before 1984 are summed up in the books

by Gilbarg

^and

Trudinger [29]g3

^and

Krylov [44]85 .

^{After the}

breakthrough

^made

in the _papers

by

^{Evans and the} author usual

technique

^{allowed to}

develop

^a

theory

which contains the

general theory

^of

quasilinear elliptic

^and

parabolic equations.

^In

particular,

the famous

Ladyzhenskaya-Ural’tseva

theorem has been

generalized

^for

fully

^nonlinear

equations.

^{It is to be mentioned that in all these}

works the data were assumed to be smoother than in linear

theory,

^{so that if}

equation ( 1.1 )

^is

just

^{Au +}

f

^{= 0 and we want to}

get

^its

solvability

^{from the}

general theory

^of

fully

^nonlinear

equations,

^{then we should assume that}

f

^is

smooth

enough.

A

major

step ^{forward in the}

general theory

has been done

by

Safonov in 1984 who

by using

^an

entirely

^new

technique proved

^the

solvability

ⁱⁿ

^C2+,

^for

(1.1)

under

only

natural smoothness

assumptions

^{on F}

(see [76]84

^and

[78]88 ).

^This

is an

extremely

strong result which is

sharp

^{even for linear}

equations.

^What

is even more

surprising,

^Safonov’s

proof

^of estimates for

(1.1)

goes the

same way when the

equation

^is

linear,

and in this very ^case it is much easier and shorter than the known

proofs

of these estimates for linear

equations.

^We

present

^this

proof

in Section 7.

The above mentioned works deal with the Holder space

theory

^of

fully

^non-

linear

elliptic equations.

^{The first}

breakthrough

^{in the Sobolev} space

theory

^has

been done

by

Caffarelli

[9]g9 (also

^{see the book}

by

Caffarelli and Cabre

[13]95)

for the

elliptic

^{case and}

Wang [98]92

^{for the}

parabolic equations.

^{The works}

by Safonov,

Caffarelli and

Wang

^are remarkable in one more

respect-they

^do

not suppose that ^{F is} convex or concave in

D2u.

But in the

general

^case

they only

^{show that to} prove ^a

priori

estimates it suffices to prove the interior

C2-estimates

for "harmonic" functions.

So far we were

talking

^about the Dirichlet

problem.

^Nonlinear

oblique

derivative

problems

^were

investigated

^{as well. An}

example

^{of such conditions}

is the

following capillarity boundary

^condition

The most relevant references here are Lions and

Trudinger [63]85 ,

^Liberman

and

Trudinger [60]86 ,

Anulova and Safonov

[5]86 . Again

^{as in the case of the}

Dirichlet

problem,

the results for

fully

^nonlinear

equations

^{contain those for}

quasilinear equations.

4. - General

degenerate fully

^nonlinear

elliptic equation

Fully

^nonlinear

degenerate elliptic equations

^{arise in}

applications

^{much more}

often than

uniformly elliptic

ones,

though

^their

investigation

^{in classes}

^C2+a heavily

^{relies on} results from the

theory

^{of nonlinear}

uniformly elliptic equations.

There is a substantial difference in difficulties which arise when we are

dealing

^with

degenerate equations

ⁱⁿ the whole space ^{or in} bounded domains.

The

theory

^{in the whole} space has been

developed mostly by probabilistic

^means

and is understood to a very

good

^extent.

Many

mathematicians contributed to

the

probabilistic

version of the

theory,

^{between them are Lions} and the author.

A PDE counterpart ^{of this}

theory

^{can be found in}

[44]85 . Degenerate equations

are

important

^not

only

^{from the}

point

^{of view of}

applications.

^The

following degenerate Monge-Ampere equation

in a sense, ^even

plays

^{the main role in the}

theory

^of

uniformly elliptic equations.

By

the way, ^{as we} have

explained

^{above the}

equation

is

uniformly elliptic

^for any E &#x3E; 0.

The

theory

of nonlinear

degenerate equations

^{cannot be} easier than the

theory

of linear ones. It is worth

mentioning

^{that even in the linear}

theory

^{there are}

still _{very many} unsolved

problems. Probably

the best references

concerning

^the

linear

theory

^are

^Kohn, Nirenberg [38]6~

^and

Oleinik,

^Radkevich

[68]71.

After the book

[44]85

the first

general

^{results on}

fully

^nonlinear

degener-

ate

equations

in domains were obtained in 1985

by Caffarelli, Nirenberg

^and

Spruck [16]85,

^where

they

considered

equations

^like

where *

^{is a}

given

^{function of x,}

f

^{is a}

symmetric

function of h E

R d

and

h(u)

^{= is a vector of}

eigenvalues

^of

Equation (1.7)

^{is a}

particular

^{case of}

(4.15).

^In

[181"

the authors

apply

^their

theory

to curvatures

of the

graph

^{of u instead of}

h(u)

and prove the

solvability

^of

equations

^for

starshaped

^{surfaces. It is worth}

noticing

^that

equations

^with curvatures are

much more

complicated

^than

containing only ^D2u.

^{One can} say ^that the latter

are

linearly fully

^nonlinear

equations

whereas the former are

quasilinearly fully

nonlinear

equations.

^{In both cases one deals with}

equation

^like

(1.3)

^{but in the}

case of

linearly fully

^nonlinear

theory

one assumes that a is

independent

^of

Du, ^{u and b is linear at} least with respect ^{to Du.}

A

general

theorem has been announced

by

^{the author in 1986}

(see [45]g6)

and

proved

^{in PDE terms in}

[48]93

^and

[51 ]95 .

^{One of its versions is}

presented

above in Section 2. This theorem allows one to consider a very

large

^{class of}

(linearly) fully

^nonlinear

degenerate elliptic equations.

Between them are the

equations

The first one is the heat

equation

^{which is a}

particular

^{case of}

degenerate fully

nonlinear

elliptic equations.

^The

general

^theorem

implies

^{that if S2 =}

{(t, x)

^E

+ r2 }

^{and r}

(3 - ~)d,

^then for any

C4-function

_g on the

boundary

^{there exists a}

unique

^{solution u E such that u =} g ^on a 0.

This result turned out to be unknown in the

theory

^{of linear}

degenerate equations.

Also,

^{in what concerns the} restrictions on r and _g, it is

sharp

^as

Weinberger’s example

^from

[38]6~

^{shows. For the second}

equation

the theorem _says that in any

C4 strictly

^{convex domain Q} and any

C4-function

_g the

equation

^{has a}

unique

^convex solution of class ¹ such that u ⁼ g ^on The

examples

^from

Caffarelli, Nirenberg

^and

Spruck [17]g6

show that all the above conditions are

necessary. In

particular,

^an

example by

^{Urbas shows that}

generally

^{the solution}

is not better than ¹ even for

analytic boundary

^{data in a ball. The same}

is true for the third

equation only

^{we have to}

speak

^about

plurisubharmonic

^u

and

strictly pseudoconvex

^{domain Q. The}

example by

^{Urbas admits an} easy

complexification.

One more

example

^of

applications

^{of the}

general

theorem from

[48] 93

and

[51 ]95

^{is the}

following equation

where

Pk (A)

^{are the k-th}

elementary symmetric polynomials

^of

eigenvalues

^of

the matrix A.

As in the case of linear

theory,

^the

theory

^{of nonlinear}

degenerate equations

is rather far from

being

^well

developed.

^{One of} very

important questions

^is

interior

regularity

of solutions. The

point

is that in many

examples

^of

equations

of

Monge-Ampere

type ^{even if one does not} have solutions

regular

up ^to the